# Properties

 Label 11.12.a.a Level $11$ Weight $12$ Character orbit 11.a Self dual yes Analytic conductor $8.452$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [11,12,Mod(1,11)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(11, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("11.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$11$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 11.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$8.45177498616$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.202533.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 37x - 2$$ x^3 - x^2 - 37*x - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}\cdot 3$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + ( - 4 \beta_{2} - 2 \beta_1 - 131) q^{3} + (17 \beta_{2} + 36 \beta_1 + 952) q^{4} + (90 \beta_{2} + 100 \beta_1 - 2435) q^{5} + ( - 110 \beta_{2} + 251 \beta_1 + 2288) q^{6} + ( - 440 \beta_{2} + 310 \beta_1 - 1694) q^{7} + ( - 404 \beta_1 - 92224) q^{8} + (348 \beta_{2} - 420 \beta_1 - 66066) q^{9}+O(q^{10})$$ q - b1 * q^2 + (-4*b2 - 2*b1 - 131) * q^3 + (17*b2 + 36*b1 + 952) * q^4 + (90*b2 + 100*b1 - 2435) * q^5 + (-110*b2 + 251*b1 + 2288) * q^6 + (-440*b2 + 310*b1 - 1694) * q^7 + (-404*b1 - 92224) * q^8 + (348*b2 - 420*b1 - 66066) * q^9 $$q - \beta_1 q^{2} + ( - 4 \beta_{2} - 2 \beta_1 - 131) q^{3} + (17 \beta_{2} + 36 \beta_1 + 952) q^{4} + (90 \beta_{2} + 100 \beta_1 - 2435) q^{5} + ( - 110 \beta_{2} + 251 \beta_1 + 2288) q^{6} + ( - 440 \beta_{2} + 310 \beta_1 - 1694) q^{7} + ( - 404 \beta_1 - 92224) q^{8} + (348 \beta_{2} - 420 \beta_1 - 66066) q^{9} + (1540 \beta_{2} - 2245 \beta_1 - 216480) q^{10} + 161051 q^{11} + ( - 35 \beta_{2} - 5908 \beta_1 - 586792) q^{12} + (5808 \beta_{2} - 4478 \beta_1 - 811404) q^{13} + ( - 21110 \beta_{2} - 4186 \beta_1 - 1338320) q^{14} + (19750 \beta_{2} + 6410 \beta_1 - 1920055) q^{15} + ( - 27948 \beta_{2} + 33040 \beta_1 - 737696) q^{16} + ( - 16280 \beta_{2} + 58778 \beta_1 + 4037374) q^{17} + (19668 \beta_{2} + 77010 \beta_1 + 1582944) q^{18} + (142340 \beta_{2} - 66050 \beta_1 - 2863520) q^{19} + ( - 90715 \beta_{2} + 74020 \beta_1 + 13151000) q^{20} + (45716 \beta_{2} - 204662 \beta_1 + 9340474) q^{21} - 161051 \beta_1 q^{22} + ( - 231814 \beta_{2} - 612722 \beta_1 - 1045471) q^{23} + (324456 \beta_{2} + 285852 \beta_1 + 13005696) q^{24} + ( - 1013500 \beta_{2} + \cdots + 19385900) q^{25}+ \cdots + (56045748 \beta_{2} + \cdots - 10639995366) q^{99}+O(q^{100})$$ q - b1 * q^2 + (-4*b2 - 2*b1 - 131) * q^3 + (17*b2 + 36*b1 + 952) * q^4 + (90*b2 + 100*b1 - 2435) * q^5 + (-110*b2 + 251*b1 + 2288) * q^6 + (-440*b2 + 310*b1 - 1694) * q^7 + (-404*b1 - 92224) * q^8 + (348*b2 - 420*b1 - 66066) * q^9 + (1540*b2 - 2245*b1 - 216480) * q^10 + 161051 * q^11 + (-35*b2 - 5908*b1 - 586792) * q^12 + (5808*b2 - 4478*b1 - 811404) * q^13 + (-21110*b2 - 4186*b1 - 1338320) * q^14 + (19750*b2 + 6410*b1 - 1920055) * q^15 + (-27948*b2 + 33040*b1 - 737696) * q^16 + (-16280*b2 + 58778*b1 + 4037374) * q^17 + (19668*b2 + 77010*b1 + 1582944) * q^18 + (142340*b2 - 66050*b1 - 2863520) * q^19 + (-90715*b2 + 74020*b1 + 13151000) * q^20 + (45716*b2 - 204662*b1 + 9340474) * q^21 - 161051*b1 * q^22 + (-231814*b2 - 612722*b1 - 1045471) * q^23 + (324456*b2 + 285852*b1 + 13005696) * q^24 + (-1013500*b2 - 559000*b1 + 19385900) * q^25 + (285214*b2 + 902916*b1 + 18823824) * q^26 + (922824*b2 + 694854*b1 + 25048935) * q^27 + (212322*b2 + 1107456*b1 - 3562768) * q^28 + (22616*b2 - 531012*b1 - 125480608) * q^29 + (602030*b2 + 1452295*b1 - 902000) * q^30 + (-257162*b2 - 1160370*b1 - 104391631) * q^31 + (-1567808*b2 + 711024*b1 + 63819008) * q^32 + (-644204*b2 - 322102*b1 - 21097681) * q^33 + (-1585306*b2 - 5958022*b1 - 191441840) * q^34 + (2500740*b2 + 3166550*b1 - 127435110) * q^35 + (-1313826*b2 - 3731160*b1 - 77474928) * q^36 + (644254*b2 - 1758996*b1 - 151427129) * q^37 + (6247090*b2 + 3533240*b1 + 330241520) * q^38 + (2689148*b2 + 4465954*b1 - 13187900) * q^39 + (-7678000*b2 - 10129380*b1 + 137107520) * q^40 + (-10552256*b2 + 7125090*b1 - 12538152) * q^41 + (5125030*b2 - 2521234*b1 + 656410448) * q^42 + (8336680*b2 + 4371400*b1 + 54054462) * q^43 + (2737867*b2 + 5797836*b1 + 153320552) * q^44 + (-7775160*b2 - 9637500*b1 + 227078070) * q^45 + (2070970*b2 + 25885231*b1 + 1623042608) * q^46 + (2025932*b2 + 27651028*b1 - 1060832728) * q^47 + (6892612*b2 - 15090256*b1 + 645289184) * q^48 + (10622020*b2 - 16352280*b1 - 262105907) * q^49 + (-26983000*b2 + 12900100*b1 + 736472000) * q^50 + (-9504836*b2 - 27646906*b1 - 299749978) * q^51 + (-16976652*b2 - 45580424*b1 - 782314016) * q^52 + (2437860*b2 - 17764276*b1 - 1000251134) * q^53 + (21409146*b2 - 61137567*b1 - 1228181328) * q^54 + (14494590*b2 + 16105100*b1 - 392159185) * q^55 + (32050120*b2 - 30280584*b1 - 384453824) * q^56 + (2622840*b2 + 64438230*b1 - 2653062720) * q^57 + (9841380*b2 + 144325648*b1 + 1614023648) * q^58 + (-15367764*b2 + 74642142*b1 - 614406569) * q^59 + (-43463935*b2 - 71732660*b1 + 134071480) * q^60 + (-3263656*b2 - 26400556*b1 - 9364704228) * q^61 + (10468458*b2 + 149250895*b1 + 3242463664) * q^62 + (17567088*b2 - 8694420*b1 - 1476360996) * q^63 + (-11290992*b2 - 138268096*b1 - 2077196416) * q^64 + (-107454160*b2 - 126041510*b1 + 3628799460) * q^65 + (-17715610*b2 + 40423801*b1 + 368484688) * q^66 + (45578896*b2 - 204548586*b1 + 3438437499) * q^67 + (77556798*b2 + 304576960*b1 + 8134360080) * q^68 + (-60667582*b2 + 87267220*b1 + 6716662141) * q^69 + (36195290*b2 - 16569570*b1 - 7178963280) * q^70 + (104141754*b2 - 264479434*b1 + 1234357219) * q^71 + (-24148080*b2 + 69846120*b1 + 6732380160) * q^72 + (197320288*b2 - 41847958*b1 + 4672344996) * q^73 + (53096076*b2 + 207019937*b1 + 5874855712) * q^74 + (-127885100*b2 - 198458800*b1 + 21376975100) * q^75 + (-126682160*b2 - 397132840*b1 + 1062068480) * q^76 + (-70862440*b2 + 49925810*b1 - 272820394) * q^77 + (20888110*b2 - 179856220*b1 - 10902332656) * q^78 + (-296828884*b2 + 353775960*b1 - 2701527686) * q^79 + (81575780*b2 + 168093200*b1 - 3670292000) * q^80 + (-95560020*b2 + 123051420*b1 - 13780039599) * q^81 + (-501007746*b2 - 117338016*b1 - 31167763568) * q^82 + (476487000*b2 + 725395500*b1 + 8669675982) * q^83 + (133735690*b2 - 207998608*b1 - 6809560912) * q^84 + (388677740*b2 + 633374010*b1 - 4457489850) * q^85 + (225806680*b2 - 311465022*b1 - 5377760960) * q^86 + (443262336*b2 + 390939564*b1 + 17147764128) * q^87 + (-65064604*b1 - 14852767424) * q^88 + (-788014172*b2 - 754367724*b1 - 5781465017) * q^89 + (-116068260*b2 + 213173850*b1 + 21697151520) * q^90 + (218497388*b2 - 44232788*b1 - 21745108984) * q^91 + (109261065*b2 - 1324907908*b1 - 73592708232) * q^92 + (292754606*b2 + 423916180*b1 + 22074200653) * q^93 + (-397133924*b2 + 41084536*b1 - 81073019104) * q^94 + (-1168748900*b2 - 1288674250*b1 + 56943524000) * q^95 + (-159817504*b2 - 770176208*b1 + 25031446528) * q^96 + (115608236*b2 + 612259096*b1 - 2661515079) * q^97 + (660381480*b2 + 723323747*b1 + 58914074560) * q^98 + (56045748*b2 - 67641420*b1 - 10639995366) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 393 q^{3} + 2856 q^{4} - 7305 q^{5} + 6864 q^{6} - 5082 q^{7} - 276672 q^{8} - 198198 q^{9}+O(q^{10})$$ 3 * q - 393 * q^3 + 2856 * q^4 - 7305 * q^5 + 6864 * q^6 - 5082 * q^7 - 276672 * q^8 - 198198 * q^9 $$3 q - 393 q^{3} + 2856 q^{4} - 7305 q^{5} + 6864 q^{6} - 5082 q^{7} - 276672 q^{8} - 198198 q^{9} - 649440 q^{10} + 483153 q^{11} - 1760376 q^{12} - 2434212 q^{13} - 4014960 q^{14} - 5760165 q^{15} - 2213088 q^{16} + 12112122 q^{17} + 4748832 q^{18} - 8590560 q^{19} + 39453000 q^{20} + 28021422 q^{21} - 3136413 q^{23} + 39017088 q^{24} + 58157700 q^{25} + 56471472 q^{26} + 75146805 q^{27} - 10688304 q^{28} - 376441824 q^{29} - 2706000 q^{30} - 313174893 q^{31} + 191457024 q^{32} - 63293043 q^{33} - 574325520 q^{34} - 382305330 q^{35} - 232424784 q^{36} - 454281387 q^{37} + 990724560 q^{38} - 39563700 q^{39} + 411322560 q^{40} - 37614456 q^{41} + 1969231344 q^{42} + 162163386 q^{43} + 459961656 q^{44} + 681234210 q^{45} + 4869127824 q^{46} - 3182498184 q^{47} + 1935867552 q^{48} - 786317721 q^{49} + 2209416000 q^{50} - 899249934 q^{51} - 2346942048 q^{52} - 3000753402 q^{53} - 3684543984 q^{54} - 1176477555 q^{55} - 1153361472 q^{56} - 7959188160 q^{57} + 4842070944 q^{58} - 1843219707 q^{59} + 402214440 q^{60} - 28094112684 q^{61} + 9727390992 q^{62} - 4429082988 q^{63} - 6231589248 q^{64} + 10886398380 q^{65} + 1105454064 q^{66} + 10315312497 q^{67} + 24403080240 q^{68} + 20149986423 q^{69} - 21536889840 q^{70} + 3703071657 q^{71} + 20197140480 q^{72} + 14017034988 q^{73} + 17624567136 q^{74} + 64130925300 q^{75} + 3186205440 q^{76} - 818461182 q^{77} - 32706997968 q^{78} - 8104583058 q^{79} - 11010876000 q^{80} - 41340118797 q^{81} - 93503290704 q^{82} + 26009027946 q^{83} - 20428682736 q^{84} - 13372469550 q^{85} - 16133282880 q^{86} + 51443292384 q^{87} - 44558302272 q^{88} - 17344395051 q^{89} + 65091454560 q^{90} - 65235326952 q^{91} - 220778124696 q^{92} + 66222601959 q^{93} - 243219057312 q^{94} + 170830572000 q^{95} + 75094339584 q^{96} - 7984545237 q^{97} + 176742223680 q^{98} - 31919986098 q^{99}+O(q^{100})$$ 3 * q - 393 * q^3 + 2856 * q^4 - 7305 * q^5 + 6864 * q^6 - 5082 * q^7 - 276672 * q^8 - 198198 * q^9 - 649440 * q^10 + 483153 * q^11 - 1760376 * q^12 - 2434212 * q^13 - 4014960 * q^14 - 5760165 * q^15 - 2213088 * q^16 + 12112122 * q^17 + 4748832 * q^18 - 8590560 * q^19 + 39453000 * q^20 + 28021422 * q^21 - 3136413 * q^23 + 39017088 * q^24 + 58157700 * q^25 + 56471472 * q^26 + 75146805 * q^27 - 10688304 * q^28 - 376441824 * q^29 - 2706000 * q^30 - 313174893 * q^31 + 191457024 * q^32 - 63293043 * q^33 - 574325520 * q^34 - 382305330 * q^35 - 232424784 * q^36 - 454281387 * q^37 + 990724560 * q^38 - 39563700 * q^39 + 411322560 * q^40 - 37614456 * q^41 + 1969231344 * q^42 + 162163386 * q^43 + 459961656 * q^44 + 681234210 * q^45 + 4869127824 * q^46 - 3182498184 * q^47 + 1935867552 * q^48 - 786317721 * q^49 + 2209416000 * q^50 - 899249934 * q^51 - 2346942048 * q^52 - 3000753402 * q^53 - 3684543984 * q^54 - 1176477555 * q^55 - 1153361472 * q^56 - 7959188160 * q^57 + 4842070944 * q^58 - 1843219707 * q^59 + 402214440 * q^60 - 28094112684 * q^61 + 9727390992 * q^62 - 4429082988 * q^63 - 6231589248 * q^64 + 10886398380 * q^65 + 1105454064 * q^66 + 10315312497 * q^67 + 24403080240 * q^68 + 20149986423 * q^69 - 21536889840 * q^70 + 3703071657 * q^71 + 20197140480 * q^72 + 14017034988 * q^73 + 17624567136 * q^74 + 64130925300 * q^75 + 3186205440 * q^76 - 818461182 * q^77 - 32706997968 * q^78 - 8104583058 * q^79 - 11010876000 * q^80 - 41340118797 * q^81 - 93503290704 * q^82 + 26009027946 * q^83 - 20428682736 * q^84 - 13372469550 * q^85 - 16133282880 * q^86 + 51443292384 * q^87 - 44558302272 * q^88 - 17344395051 * q^89 + 65091454560 * q^90 - 65235326952 * q^91 - 220778124696 * q^92 + 66222601959 * q^93 - 243219057312 * q^94 + 170830572000 * q^95 + 75094339584 * q^96 - 7984545237 * q^97 + 176742223680 * q^98 - 31919986098 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 37x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu^{2} + 6\nu - 52$$ 2*v^2 + 6*v - 52 $$\beta_{2}$$ $$=$$ $$-4\nu^{2} + 12\nu + 96$$ -4*v^2 + 12*v + 96
 $$\nu$$ $$=$$ $$( \beta_{2} + 2\beta _1 + 8 ) / 24$$ (b2 + 2*b1 + 8) / 24 $$\nu^{2}$$ $$=$$ $$( -\beta_{2} + 2\beta _1 + 200 ) / 8$$ (-b2 + 2*b1 + 200) / 8

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 6.62795 −5.57381 −0.0541376
−75.6271 −281.520 3671.46 5111.20 21290.6 21831.1 −122777. −97893.2 −386545.
1.2 23.3081 296.237 −1504.73 −13329.8 6904.73 32948.8 −82807.5 −89390.6 −310692.
1.3 52.3190 −407.717 689.274 913.580 −21331.3 −59861.9 −71087.1 −10914.2 47797.6
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.12.a.a 3
3.b odd 2 1 99.12.a.a 3
4.b odd 2 1 176.12.a.e 3
5.b even 2 1 275.12.a.a 3
11.b odd 2 1 121.12.a.c 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.12.a.a 3 1.a even 1 1 trivial
99.12.a.a 3 3.b odd 2 1
121.12.a.c 3 11.b odd 2 1
176.12.a.e 3 4.b odd 2 1
275.12.a.a 3 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 4500T_{2} + 92224$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(11))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 4500T + 92224$$
$3$ $$T^{3} + 393 T^{2} + \cdots - 34002261$$
$5$ $$T^{3} + \cdots + 62243294875$$
$7$ $$T^{3} + \cdots + 43059126844184$$
$11$ $$(T - 161051)^{3}$$
$13$ $$T^{3} + \cdots + 33\!\cdots\!56$$
$17$ $$T^{3} + \cdots + 21\!\cdots\!48$$
$19$ $$T^{3} + \cdots - 16\!\cdots\!00$$
$23$ $$T^{3} + \cdots + 14\!\cdots\!59$$
$29$ $$T^{3} + \cdots + 18\!\cdots\!40$$
$31$ $$T^{3} + \cdots + 69\!\cdots\!75$$
$37$ $$T^{3} + \cdots - 98\!\cdots\!23$$
$41$ $$T^{3} + \cdots + 60\!\cdots\!08$$
$43$ $$T^{3} + \cdots + 19\!\cdots\!72$$
$47$ $$T^{3} + \cdots - 45\!\cdots\!28$$
$53$ $$T^{3} + \cdots - 30\!\cdots\!72$$
$59$ $$T^{3} + \cdots - 26\!\cdots\!15$$
$61$ $$T^{3} + \cdots + 79\!\cdots\!68$$
$67$ $$T^{3} + \cdots + 86\!\cdots\!09$$
$71$ $$T^{3} + \cdots - 11\!\cdots\!39$$
$73$ $$T^{3} + \cdots + 49\!\cdots\!76$$
$79$ $$T^{3} + \cdots + 20\!\cdots\!20$$
$83$ $$T^{3} + \cdots + 54\!\cdots\!32$$
$89$ $$T^{3} + \cdots - 22\!\cdots\!15$$
$97$ $$T^{3} + \cdots - 28\!\cdots\!93$$